Complex Numbers
Complex Numbers
star_batch_jee_advanced_2025
Grade 11
Question:
$a_1, a_2, a_3, \ldots, a_{100}$ are all the $100^{th}$ roots of unity. The numerical value of $\sum\limits_{1 \leq i 0)$, then:
Maximum area of $(\triangle PAB) = \frac{1}{2}k|z_1 - z_2|$
There are two possible positions of $P$ on argand plane when area of $\triangle PAB$ is maximum
Area of $\triangle PAB = \text{constant} \left(< \frac{1}{2}k|z_1 - z_2|\right)$, for 4 possible positions of $P$.
$\triangle PAB$ is equilateral triangle of maximum area if $4k^2 = 3|z_1 - z_2|^2$
Step-by-Step Solution
Key Concept: Use multinomial coefficients to partition a set into groups of specified sizes.
Let $D_1, D_2, D_3$ be districts requiring 2, 3, 5 men respectively. The number of ways to distribute is $\frac{10!}{3!2!5!}$, which accounts for assigning 10 people to three districts with specified sizes.
Correct Answer: 2,4